Full text of DOI:10.1111/1468-0262.00110

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Econometrica, Vol. 68, No. 2 March, 2000 , 275᎐307
MORTGAGE TERMINATIONS, HETEROGENEITY AND THE
EXERCISE OF MORTGAGE OPTIONS
BY YONGHENG DENG, JOHN M. QUIGLEY, AND
1
ROBERT VAN ORDER
As applied to the behavior of homeowners with mortgages, option theory predicts that
mortgage prepayment or default will be exercised if the call or put option is ‘‘in the
money’’ by some specific amount. Our analysis: tests the extent to which the option
approach can explain default and prepayment behavior; evaluates the practical impor-
tance of modeling both options simultaneously; and models the unobserved heterogeneity
of borrowers in the home mortgage market. The paper presents a unified model of the
competing risks of mortgage termination by prepayment and default, considering the two
hazards as dependent competing risks that are estimated jointly. It also accounts for the
unobserved heterogeneity among borrowers, and estimates the unobserved heterogeneity
simultaneously with the parameters and baseline hazards associated with prepayment and
default functions.
Our results show that the option model, in its most straightforward version, does a
good job of explaining default and prepayment, but it is not enough by itself. The
simultaneity of the options is very important empirically in explaining behavior. The
results also show that there exists significant heterogeneity among mortgage borrowers.
Ignoring this heterogeneity results in serious errors in estimating the prepayment behav-
ior of homeowners.
KEYWORDS: Mortgage default, prepayment, dependent competing risks, mortgage
pricing.
1. INTRODUCTION
THE MORTGAGE MARKET IS QUITE LARGE and is increasing in importance. The
outstanding volume of residential mortgages is currently over $ 3 trillion, and
volume has doubled in the past decade. In comparison, the stock of outstanding
U.S. government debt is currently about $ 5 trillion. Almost half of the stock of
mortgages is held in ‘‘mortgage-backed securities,’’ and about half of all new
mortgages are ‘‘securitized.’’ The rise of securitization, the trading of these
securities, and the growing use of mortgage-backed securities as collateral for
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‘‘derivatives’’ e.g., collateralized mortgage obligations has generated a great
deal of interest in the economics of mortgage and mortgage-backed securities.
Pricing mortgage contracts is complicated, primarily by the options available
to the borrower to default or to prepay. These options are distinct, but not
1The authors thank Brian McCall for providing us the Fortran code used in his 1996 Economet-
rica paper. The paper benefited from the comments of Jushan Bai, Glenn Sueyoshi, a co-editor and
two anonymous referees. The views expressed in this research are those of the authors and do not
represent the policies or positions of Freddie Mac. Quigley’s research is supported by the Berkeley
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Program on Housing and Urban Policy http:rrurbanpolicy.berkeley.edu . Deng’s initial research
was completed at the University of California, Berkeley supported by the Fisher Center for Real
Estate and Urban Economics.
275

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Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
independent. Thus, one cannot calculate accurately the economic value of the
default option without considering simultaneously the financial incentive for
prepayment. Furthermore, risk preferences and other idiosyncratic differences
across borrowers may vary widely. Typically, it is very difficult to measure this
heterogeneity explicitly. Appropriately modeling these prepayment and default
risks is crucial to the pricing of mortgages and to understanding the economic
behavior of homeowners.
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The contingent claims models, developed by Black and Scholes 1973 , Mer-
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ton 1973 , Cox, Ingersoll, and Ross 1985 , and others, provide a coherent
motivation for borrower behavior, and a number of studies have applied this
model to the mortgage market. Hendershott and Van Order 1987 and Kau and
Keenan 1995 have surveyed much of the literature related to mortgage pricing.
Virtually all previous studies using option models, however, focus on applying
them to explain either prepayment or default behavior, but not both. For
instance, in the first application of option models to mortgages, Findley and
Capozza 1977 analyzed the prepayment options of holders of adjustable-rate
and fixed-rate mortgages. Dunn and McConnell 1981 , Buser and Hendershott
1984 , and Brennan and Schwartz 1985 used option theory to price callable
mortgages, relying on simulation methods. Green and Shoven 1986 , Schwartz
and Torous 1989 , and Quigley and Van Order 1990 provided empirical
estimates of option-based prepayment models.
Cunningham and Hendershott 1984 and Epperson, Kau, Keenan, and Muller
1985 applied option models to price default risk, modeling default as a put
option, and Foster and Van Order 1984 and Quigley and Van Order 1995
estimated default models empirically in an option-based framework. Quercia
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and Stegman 1992 and Vandell 1993 reviewed many of these default models.
A series of papers by Kau, Keenan, Muller, and Epperson 1992, 1995 , Kau
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and Keenan 1996 , and Titman and Torous 1989 provided theoretical models
that emphasized the importance of the jointness of prepayment and default
options. A homeowner who exercises the default option today gives up the
option to default in the future, but she also gives up the option to prepay the
mortgage. Foster and Van Order 1985 estimated simultaneous models of
default and prepayment using data on large pools of FHA loans, and Schwartz
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and Torous 1993 estimated the joint hazard using a Poisson regression ap-
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proach and aggregate data. Deng, Quigley, and Van Order 1996 and Deng
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1997 were the first to analyze residential mortgage prepayment and default
behavior using micro data on the joint choices of individuals. However, the
competing risks hazard model common to all these studies ignores the hetero-
geneity among borrowers. Presumably a substantial number of homeowners are
less likely to exercise put and call options on mortgages in the fully rational way
predicted by finance theory. Accounting for this group is potentially important
in understanding market behavior and in pricing seasoned mortgages.
In this paper, we present a unified economic model of the competing risks of
mortgage termination by prepayment and default. We adopt a proportional
hazard framework to analyze these competing risks empirically, using a large
sample of individual loans, and we extend the model to analyze unobserved

MORTGAGE TERMINATIONS
277
heterogeneity. We test three aspects of homeowner behavior in the mortgage
market:
1. the extent to which the option approach can explain the default and
prepayment behavior of borrowers with single-family mortgages;
2. the importance of modeling both options simultaneously; and
3. the importance of heterogeneity of borrowers in explaining behavior in the
market.
We find that:
1. The option model, in its most straightforward version, does a good job of
explaining default and prepayment, but it is not enough by itself. Either
transactions costs vary a great deal across borrowers, or else some people are
simply much worse at exercising options.
2. The simultaneity of the options is very important empirically in explaining
behavior. In particular, factors that trigger one option are also important in
triggering or foregoing exercise of the other.
3. Unobserved borrower heterogeneity is quite important in accounting for
borrower behavior. We allow for heterogeneity by incorporating into the estima-
tion the possibility that there are different sorts of borrowers, some very astute,
some quite passive, and others somewhere in between.2 We find that hetero-
geneity is significant. It has important effects on key elasticities explaining
behavior, particularly with respect to prepayment.
The paper is organized as follows: Section 2 reviews the application of option
models to mortgage terminations. Section 3 discusses the proportional hazard
model, specified with competing risks, time-varying covariates and unobserved
heterogeneity. Section 4 presents an extensive empirical analysis. Section 5 is a
brief conclusion.
2. MORTGAGE TERMINATIONS AND OPTION PRICING
Well-informed borrowers in a perfectly competitive market will exercise
financial options when they can thereby increase their wealth. In the absence of
either transactions costs or reputation costs which reduce credit ratings, and
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with assumable mortgages or no exogenous reasons for residential mobility ,
default and prepayment are essentially financial decisions which can be sepa-
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rated from real housing decisions, and the simplest version of the Miller and
Modigliani theory of the irrelevance of financial structure holds.3 Under these
conditions, individuals can increase their wealth by defaulting on a mortgage
when the market value of the mortgage equals or exceeds the value of the
house. Similarly, by prepaying the mortgage when market value equals or
exceeds par, they can increase wealth by refinancing. A necessary condition for
exercising an option is that it be ‘‘in the money,’’ but that is not sufficient.
Exercising either option now means giving up the option to exercise both
options later. For instance, a borrower whose house price declines below the
2
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The unobserved heterogeneity may be attributed to unmeasured house-specific factors such as
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unexpected depreciation or appreciation of property values as well as to borrower tastes or abilities.
3
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See Kau, Keenan, Muller, and Epperson 1995 for a recent discussion.

278
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
mortgage balance may not default immediately, in part because after the price
decline the mortgage has a below-market rate,4 but also because by defaulting,
the borrower would also lose the option to refinance later on.
While virtually all the recent research on prepayment and default, summa-
rized above, has used option-based models, the underlying theories behind the
models differ importantly in the treatment of transactions costs.5 For simplicity,
we divide these approaches into polar cases. The first case, Model I, assumes no
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transactions costs see Titman and Torous 1989 , Kau, Keenan, Muller, and
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Epperson 1992 , and ‘‘ruthless’’ exercise of both options. The second case,
Model II, emphasizes transactions costs, particularly in exercise of the default
option. It is assumed that transactions costs are sufficiently high that default
requires, not only negative equity, but also a ‘‘trigger event’’ that forces the
borrower to leave the house. Model II also entertains the possibility that there
are significant transactions costs involved in prepaying, or else that some
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borrowers are more astute than others at exercising options see Archer, Ling,
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and McGill 1996 . Finally, Model II also allows the possibility of significant
unobserved heterogeneity. Thus the separation between housing and finance
decisions is incomplete.
According to Model I, understanding when to exercise either option requires
specifying the underlying state variables and the parameters that determine the
value of the contract and then deducing the rule for exercise that maximizes
borrower wealth. For residential mortgages, the key state variables are interest
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rates and house values. The value of a mortgage M c, r, H, B, k depends upon
the coupon rate, c, a vector of relevant interest rates, r, property value, H, the
outstanding balance, B, the age of the loan, k, and some other parameters. With
continuous time, a standard arbitrage argument is sufficient to derive an
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equilibrium condition for M a second order partial differential equation such
that the value of the mortgage equals the risk-adjusted expected present value
of its net cash flows.
Assume that house price changes are continuous with an instantaneous mean
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␮ and a standard deviation ␴h. Let d be the imputed rent payout ‘‘dividend’’
rate. For simplicity, assume there is only one interest rate, the instantaneous
short rate r, which determines the yield curve. Let ␪ be the mean value of the
short rate, ␥ be the rate of convergence for the short rate, ␴ be the volatility of
r
the short rate, and ␳ be the correlation between interest rate changes and
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house price changes. Then it has been shown Kau, Keenan, Muller, and
..
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Epperson 1995 that the value of the mortgage M satisfies
1
2
2 Ѩ 2M
Ѩ r2
Ѩ 2M
h Ѩ r ѨH
1
2
2 Ѩ 2M
ѨH 2
ѨM
Ѩ r
r␴
q␳ r H␴ ␴
q
H 2␴
h
q␥ ␪yr
'
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1
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.
r
r
ѨM
ѨM
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.
q ryd H
q
yrMs0.
ѨH
Ѩ␶
4 This is because the mortgage is now riskier, so defaulting and buying back the same house would
require paying a higher interest rate or making a larger downpayment.
5
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See Kau, Keenan, and Kim 1993 for an explicit discussion of transaction costs.

MORTGAGE TERMINATIONS
279
Ž .
The value of M и and the optimal default and prepayment strategy are
Ž .
determined simultaneously. Equation 1 is consistent with an infinite number of
Ž .
functions M и . The appropriate function is determined by choosing the optimal
level of r, rU, and the optimal level of H, HU, at which to terminate the
mortgage through default or prepayment. These are the levels of r and H that
Ž . Ž
1 see Kau, Keenan, Muller, and Epperson
minimize M given equation
1995 ; these levels are functions of c, d, B, k and the parameters governing the
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..
stochastic processes for r and H. Due to the jointness of the options, there are
two pairs of r and H that trigger termination. There are levels of r that trigger
default as well as prepayment, and levels of H that trigger prepayment as well
as default. For instance, a borrower may default at a low enough level of r as a
means of prepayment, and a borrower might refinance when equity value has
risen because the loan is now safer and would carry a lower interest rate. The
estimated probability of default or prepayment is the probability of these levels
of r and H occurring, conditional on the information set of actors in the market
and the researchers observing them.
Ž .
Note that the borrower does not have to solve 1 and the boundary condi-
tions in order to know when to exercise either option. All that is necessary is
knowledge of market prices. For instance, for a fixed-rate mortgage, the prepay-
ment option should be exercised whenever the borrower can refinance the loan
for the same remaining term at par at a mortgage rate less than the coupon on
the current loan or, alternatively when the market value of the mortgage equals
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or exceeds the mortgage balance. Default should be exercised when the
borrower’s payments would be lower on a new zero-downpayment loan for the
same remaining term, used to purchase the same house. Of course, we on the
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outside do not observe these market alternatives and markets are not that
.
complete anyway ; this greatly complicates testing the model.
Due to ‘‘data limitations,’’ the analyst does not observe the critical levels of
house price and the mortgage rate that trigger exercise from the details of the
mortgage contract. All that we can hope to observe is the extent to which either
option is ‘‘in the money.’’ But we cannot even measure directly the extent to
which the default option is in the money without data on the course of
individual house prices. We can however estimate the probability that the option
is in the money, given the initial loan-to-value ratio and the stochastic process
for house prices. The analyst can control for the remaining term of the loan, but
not for changes in the parameters of the house price or interest rate process.
This reality suggests that it is more productive to consider optimal exercise in
probabilistic terms and then to test some of the major predictions of Model I:
First, the probability of exercise should increase as the option moves further
into the money. Second, the probability of exercise should accelerate as the
option moves further into the money. Third, because exercising one option
means giving up the other option, the extent to which one option is ‘‘in the
money’’ should affect the exercise of the other. Thus, for example, the probabil-
ity of prepayment is a function of the extent to which the default option is in the
money.

280
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
Model I can be extended to address asymmetric information. For instance, we
cannot observe directly the parameters governing house price volatilities. This
can be a problem if the volatilities vary in a systematic way, for instance if
borrowers know more about their own house price volatility than lenders do.
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Then risky houses might be financed with high loan-to-value LTV loans, as
borrowers exploit underpriced options.6 One may control for this by using initial
LTV as an explanatory variable in predicting defaults. Similarly, borrowers who
expect to move sooner than average will choose to pay fewer ‘‘points’’ up-front;
as a result they will have higher than average coupons. Also lenders may charge
higher rates to borrowers with riskier houses, so that high rates will be
associated with higher defaults.
Model I has the great advantage of simplicity.
Model II incorporates transactions costs in a broad sense. It is not simple
because transaction costs are complicated and are generally not observable. For
instance, different transaction costs across borrowers have been used to explain
the observation that the prepayments in mortgage pools tend to be slower than
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expected and drawn out over time e.g., see Archer and Ling 1993 , Stanton
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1995 , Harding 1997 . This raises the general question of unobserved differ-
ences among mortgage borrowers. Whether this empirical finding arises from
variations in transaction costs or differences in the astuteness of homeowners
exercising options, unobserved heterogeneity means that surviving borrowers are
systematically different over time. For instance, surviving borrowers may be
increasingly less interest-rate sensitive over time if more astute borrowers
refinance first, something of obvious importance in pricing seasoned mortgages.
Transactions costs are more complicated on the default side, particularly if
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the mortgage is not assumable. A borrower forced to move e.g., due to divorce
.
or job loss who cannot have the mortgage assumed has a very short remaining
term and may thus default with little negative equity. On the other hand, if
there are costs to defaulting, HU may be lower than Model I implies. For these
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reasons, many researchers see Quigley and Van Order 1995 for a discussion
estimate modified option models, which predict that exercise is a function of
both ‘‘trigger events’’ like default or divorce and also the extent to which the
option is in the money.
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We follow Kau and Keenan 1996 who introduce random terminations into
the model. These terminations force either a prepayment or a default. If
mortgages are not assumable7 and there are no e.g., reputation costs to
default, a random termination will lead to default if the house is worth less than
the mortgage balance, and prepayment otherwise. Note that in Model II, it is
the par value of the mortgage that is relevant for default. In contrast, in Model I
a borrower is less likely to default when interest rates increase due to the value
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6
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See Yezer, Phillips, and Trost 1994 for a discussion.
7The empirical analysis below is based on mortgages that were nominally not assumable, but
some states forbade exercise of due-on-sale clauses during the observation period, and in any event
due on sales clauses were typically not enforced.

MORTGAGE TERMINATIONS
281
of the low-rate mortgage. According to Model II, a borrower who is forced to
leave the house does not have the option to keep the mortgage alive. As is the
case with prepayments, transactions costs matter, especially if they vary across
borrowers or if there are unobservable differences in astuteness among borrow-
ers.
Estimates of default and prepayment are reported below in three stages. First,
we estimate proportional hazard models that use as explanatory variables only
the extent to which the options are ‘‘in the money,’’ in order to test the
predictions of Model I. Second, we add variables that are proxies for informa-
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tion asymmetry e.g., the original loan-to-value ratio and trigger events e.g.,
.
unemployment and divorce . Finally, we allow for unobserved heterogeneity and
estimate the nonparametric distribution of the unobserved heterogeneity simul-
taneously with the competing risks of prepayment and default functions.
3. A COMPETING RISKS MODEL OF MORTGAGE TERMINATION WITH
UNOBSERVED HETEROGENEITY
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The proportional hazard model introduced by Cox Cox and Oakes 1984 ,
provides a convenient framework for considering the exercise of options empiri-
cally and the importance of other trigger events in mortgage terminations.
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. Ž
Han and Hausman 1990 , Sueyoshi 1992 , and McCall 1996 HHSM, for
short suggested a maximum likelihood estimation approach for the proportional
.
hazard model with grouped duration data. The HHSM approach estimates the
competing risks simultaneously, accounts for the fact that risks may be corre-
lated, and also that covariates may be time-varying. There is no restriction on
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the functional form of the baseline hazard described below . Following HHSM,
the competing risks model for mortgage prepayment and default can be derived:
Let Tp and Td be the discrete random variables representing the duration of a
mortgage until it is terminated by the mortgage holder in the form of prepay-
ment or default, respectively. The joint survivor function conditional on ␩p, ␩d,
r, H, Y, and X can be expressed in the following form:
Ž .
2
<
S t , t r, H, Y, X, ␩ , ␩
Ž
.
p
d
p
d
tp
X
p
Ž .
exp ␥ qg r, H, Y q␤ X
sexp y␩
Ž
.
Ý
p
pk
pk
ž
k
s1
td
X
Ž
Ž
dk
.
.
y␩
exp ␥ qg r, H, Y q␤d X
,
d Ý
dk
/
ks1
8
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.
where gjk r, H, Y are time-varying functions of options-related variables, r and
H are the relevant interest rates and property values, respectively, as discussed
in the previous section; Y is a vector of other variables that will be used,
8 The details of the function gjk r, H, Y are specified in the following section and the Appendix.
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282
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
together with r and H, to estimate the market values of the options empirically;
X is a vector of other non-option-related variables, which may include indicators
reflecting a borrower’s credit risk or financial strength, as well as other trigger
events, such as unemployment and divorce. To simplify the notation, we sup-
press the time-varying subscripts for r, H, Y, and X. ␥jk are parameters of the
baseline function which may be estimated nonparametrically, following Han and
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.
Hausman 1990 :
k
Ž .
3
Ž .
h0 j t dt , jsp, d.
␥ slog
H
jk
ky1
␩ and ␩ are unobserved heterogeneities associated with the hazard functions
p
for prepadyment and default respectively.
We allow for the possibility that the population of mortgage borrowers
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.
consists of M distinct groups. The joint distribution of the unobservables ␩p, ␩
d
is modeled by assuming that these distinct, but unobserved types of individuals,
Ž
ms1, 2, . . . , M an individual in group m is characterized by the doublet of
Ž
..
, occur in the population with relative frequency
location parameters ␩pm, ␩
dm
pm, ms1, 2, . . . , M.
Due to the nature of the competing risks between prepayment and default,
only the duration associated with the type that terminates first is observed, i.e.
Ž
.
Ž <
.
tsmin tp, td . Define Fp k ␩p, ␩ as the probability of mortgage termination by
d
Ž <
.
prepayment in period k, Fd k ␩d, ␩ as the probability of mortgage termination
by default in period k, Fu k ␩p, ␩ as the probability of mortgage termination in
period k but information on tdhe cause of the termination is missing, and
d
Ž <
.
Ž <
.
Fc k ␩p, ␩ as the probability that mortgage duration data are censored in
d
period k due to the ending of the data collecting period.
Ž
.
Following McCall 1996 , these probabilities can be expressed as
1
Ž .
<
<
<
<
4
F
k ␩ , ␩ sS k, k ␩ , ␩ yS kq1, k ␩ , ␩
y
S k, k ␩ , ␩
p Ž
.
Ž
.
Ž
.
Ž
.
.
p
d
p
d
p
d
p
d
2
<
<
qS kq1, kq1 ␩ , ␩ yS k, kq1 ␩ , ␩
Ž
.
Ž
.
p
d
p
d
<
yS kq1, k ␩ , ␩
,
Ž
.
p
d
1
2
Ž .
5
<
<
<
<
Fd k ␩ , ␩ sS k, k ␩ , ␩ yS k, kq1 ␩ , ␩
y
S k, k ␩ , ␩
Ž
.
Ž
.
Ž
.
Ž
p
d
p
d
p
d
p
d
<
<
qS kq1, kq1 ␩ , ␩ yS k, kq1 ␩ , ␩
Ž
.
Ž
.
p
d
p
d
<
yS kq1, k ␩ , ␩
,
Ž
.
p
d
Ž .
6
<
<
<
p
F k ␩ , ␩ sS k, k ␩ , ␩ yS kq1, kq1 ␩ , ␩
,
d
Ž
.
Ž
.
Ž
.
u
p
d
p
d

MORTGAGE TERMINATIONS
283
and
Ž .
<
<
p
7
F k ␩ , ␩ sS k, k ␩ , ␩
c Ž .
,
.
Ž
p
d
d
where the dependence of these functions on r, H, Y, and X has been omitted
for notational simplicity.
The unconditional probability is given by
M
Ž .
8
Ž .
Fj k s
<
p F k ␩pm , ␩
,
jsp, d, u, c.
j Ž
.
Ý
m
dm
m
s1
The log likelihood function of the competing risks model is given by
N
Ž .
9
Ž
.
Ž
Ž
.. ..
Ž
Ž
log Ls
␦
pi
log F Ki q␦ log Fd Ki q␦ui log Fu Ki
Ž
.
Ý
p
di
i
s1
Ž
Ž
..
q␦ci log Fc Ki ,
where ␦ji, jsp, d, u, c are indicator variables that take value one if the ith loan
is terminated by prepayment, default, unknown type, or censoring, respectively,
and take a value of zero otherwise.
4. THE EMPIRICAL ANALYSIS
The empirical analysis is based upon individual mortgage history data main-
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tained by the Federal Home Loan Mortgage Corporation Freddie Mac . The
data base contains 1,489,372 observations on single family mortgage loans issued
between 1976 to 1983 and purchased by Freddie Mac. All are fixed-rate,
level-payment, fully-amortized loans, most of them with thirty-year terms. The
mortgage history period ends in the first quarter of 1992. For each mortgage
loan, the available information includes the year and month of origination and
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termination if it has been closed , indicators of prepayment or default, the
purchase price of the property, the original loan amount, the initial loan-to-value
ratio, the mortgage contract interest rate, the monthly principal and interest
payment, the state, the region, and the major metropolitan area in which the
property is located. For the mortgage default and prepayment model, censored
observations include all matured loans as well as the loans active at the end of
the period.
The analysis is confined to mortgage loans issued for owner occupancy, and
includes only those loans that were either closed or still active at the first
quarter of 1992. The analysis is confined to loans issued in 30 major metropoli-
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.
tan areas MSAs ᎏa total of 447,042 observations. Loans are observed in each
quarter from the quarter of origination through the quarter of termination,
maturation, or through 1992:I for active loans.
The key variables are those measuring the extent to which the put and call
options are in the money. To value the call option, the current mortgage interest

284
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
rate and the initial contract terms are sufficient. We compute a variable
‘‘Call Option’’ measuring the ratio of the present discounted value of the
unpaid mortgage balance at the current quarterly mortgage interest rate9
relative to the value discounted at the contract interest rate.10
To value the put option analogously, we would measure the market value of
each house quarterly and compute homeowner equity quarterly. Obviously, we
do not observe the course of price variation for individual houses in the sample.
Ž
.
We do, however, have access to a large sample of repeat or paired sales of
Ž
.
single family houses in these 30 metropolitan areas MSAs . This information is
Ž
.
sufficient to estimate a weighted repeat sales house price index WRS sepa-
rately for each of the 30 MSAs. The WRS index provides estimates of the course
of house prices in each metropolitan area. It also provides an estimate of the
variance in price for each house in the sample, by metropolitan area and elapsed
time since purchase.11
Estimates of the mean and variance of individual house prices, together with
Ž
.
the unpaid mortgage balance computed from the contract terms , permit us to
estimate the distribution of homeowner equity quarterly for each observation. In
particular, the variable ‘‘Put Option’’ measures the probability that homeowner
equity is negative, i.e., the probability that the put option is in the money.12
As proxies for other ‘‘trigger events,’’ we include measures of the quarterly
unemployment rate and the annual divorce rate by state.13
Figure 1 summarizes the raw data used in the empirical analysis described
Ž
.
below. Panel A displays the average Kaplan-Meier conditional prepayment
Ž
.
rates, separately by the loan-to-value ratio at origination LTV , as a function of
duration. Conditional prepayment rates are slightly higher for higher LTV loans.
Rates increase substantially after the first fifteen quarters. Panel C of Figure 1
displays average Kaplan-Meier conditional default rates by LTV. Note again
that default rates increase substantially after about fifteen quarters. Note also
that the default rates vary substantially by initial LTV. Default rates for 90
percent LTV loans are four or five times higher than default rates for 80 to 90
percent LTV loans. The default rates for these latter loans are, in turn, about
twice as high as for those with LTV below 80 percent.
Ž
.
Finally, note that conditional default rates are quite low. Even for the riskiest
class of loans, conditional default rates are no higher than two in a thousand in
9 The rate used is the average interest rate charged by lenders on new first mortgages reported by
Freddie Mac’s market survey. This mortgage interest rate varies by quarter across five major US
regions.
10 See Appendix A for the specification of the ‘‘Call Option’’ variable.
11 See Appendix B for the specification of the house price indices and their volatilities.
12 See Appendix A for the specification of the ‘‘Put Option’’ variable.
13State unemployment data are reported in various issues of: US Department of Labor, ‘‘Employ-
Ž
.
ment and Unemployment in States and Local Areas Monthly ’’ and in the ‘‘Monthly Labor Re¨iew.’’
State divorce data are reported in various issues of U.S. National Center for Health Statistics, ‘‘Vital
Statistics of the United States, Volume III, Marriage and Di¨orce,’’ and in ‘‘Statistical Abstract of the
U.S.’’

MORTGAGE TERMINATIONS
285
FIGURE 1
any quarter. Residential mortgages are relatively safe investments, and the
Ž
period as a whole was one of generally rising house prices keeping the put
option out of the money .
.
Table I presents the means and standard deviations of the explanatory
variables measured at origination and termination of the mortgage loans. The

286
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
Ž
.
Continued
FIGURE 1
mean value of the prepayment option, ‘‘Call Option,’’ is ‘‘out of the money’’
when mortgages were originated, but is much less so when the mortgages were
Ž
terminated presumably reflecting the fact that most prepayments arise because
Ž
..
homeowners move for other reasons; see Quigley 1987 . The mean value of the
probability of negative equity, ‘‘Put Option,’’ is quite low in general when
mortgages were originated. The initial probability is about 0.007 for those

MORTGAGE TERMINATIONS
TABLE I
287
DESCRIPTIVE STATISTICS ON MORTGAGE LOANS MEAN VALUES AT ORIGINATION AND TERMINATION
At Origination At Termination
Variable
All Loans
Prepaid
Defaulted
Othera
Prepaid
Defaulted
Ž
Call Option fraction of y0.0529
y0.0559
y0.0746
y0.0427
y0.0265
0.0321
.
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
contract value
0.0067
0.0072
0.0073
0.0069
0.0102
0.0045
0.0329
0.0067
0.0273
0.0918
0.0283
0.0283
Ž
Put Option probability
0.0371
0.0030
0.0157
0.0062
0.0004
0.0063
.
.
.
.
.
.
.
of negative equity
Squared Term of Call
Option
Squared Term of Put
Option
State Divorce Rate
percent
State Unemployment
Rate percent
Initial Loan-To-Value
0.0005
0.0095
0.0005
0.0104
0.0013
0.0336
.
.
.
.
.
.
0.0005
0.0006
0.0006
0.0005
0.0009
0.0044
0.0004
0.0004
0.0035
0.0014
0.0021
0.0367
.
.
.
.
.
.
0.0000
5.4486
0.0000
5.4448
0.0002
5.8430
0.0000
5.4349
0.0005
4.8635
0.0162
5.2587
Ž
.
.
.
.
.
.
.
0.8015
6.8605
0.7659
6.9609
0.5083
0.9105
6.5648
0.5585
6.6796
0.3650
7.6085
6.88165
Ž
.
.
.
.
.
.
.
2.4960
0.7657
2.5356
0.7649
3.6764
0.8900
2.1951
0.7597
2.8578
ᎏ
3.5006
ᎏ
Ž
.
.
.
.
.
Ratio LTV
0.0240
0.0240
0.0060
0.0239
No. of Observations
22,294
16,402
363
5,529
16,402
363
N
OTE: Standard deviations are in parentheses.
a Other includes matured mortgages as well as those outstanding at the end of the observation period.
mortgages that were ultimately terminated by prepayment, but the initial proba-
bility of negative equity is much larger for those loans that were ultimately
terminated by default. In part, this difference reflects the much higher initial
loan-to-value ratios of those mortgages ultimately terminated by default. Finally,
at termination the defaulted mortgages are associated with substantially higher
average unemployment rates and divorce rates than those mortgages terminated
by prepayment.
Table II presents the means and standard deviations of the explanatory
variables measured at termination of the mortgage loans, separately by initial
LTV categories. The differences in the values of the call option across LTV
groups are insignificant. However, the put option values are significantly differ-
ent among different LTV groups, i.e., higher LTV ratios at origination are
associated with higher probabilities of negative equity at the termination of the
mortgage loans.
Table II also presents cumulative prepayment and default rates at different
ages of mortgage life by initial LTV categories. The differences in cumulative
prepayment rates among different LTV categories are relatively small, but a
higher initial LTV is strongly associated with a higher cumulative default rate.
This strong association persists during different stages of the mortgage life.
4.1. Specifications and Results
Table III presents maximum likelihood estimates of the parameters of models
of competing risks of mortgage prepayment and default. Estimates in this table

288
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
TABLE II
DESCRIPTIVE STATISTICS ON MORTGAGE LOANS
a
MEAN VALUES AT TERMINATION AND CUMULATIVE RATES BY INITIAL LTV CATEGORY
Variable
All Loans
LTVF 75 75- LTV F 80 80- LTV F 90 LTV G 90
A. Mean Value
Ž
Call Option fraction of y0.0057
0.0000
y0.0161
y0.0030
0.0040
.
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
contract value
0.0270
0.0067
0.0267
0.0006
0.0002
0.0267
0.0259
0.0019
0.0273
0.0105
0.0296
0.0334
0.0100
0.0297
Ž
Put Option probability
.
.
.
.
.
.
of negative equity
Squared Term of Call
Option
Squared Term of Put
Option
0.0016
0.0270
0.0001
0.0262
0.0016
0.0273
.
.
.
.
.
0.0028
0.0016
0.0028
0.0002
0.0028
0.0001
0.0028
0.0017
0.0030
0.0111
.
.
.
.
.
0.0007
7.1593
0.0000
7.2773
0.0000
7.2123
0.0005
7.1006
0.0051
6.7308
State Unemployment
Ž
.
.
.
.
.
.
Rate percent
State Divorce Rate
3.2006
4.8068
3.3740
4.7378
3.1218
4.8044
3.1085
4.8498
2.9797
4.9498
Ž
.
.
.
.
.
.
0.8902
percent
B. Cumulative Rates
Cumulative Prepayment 18.8
0.6165
0.6066
0.5152
0.5878
19.8
55.6
72.6
0.1
17.0
56.7
74.4
0.4
18.6
60.2
75.0
1.4
21.7
55.8
67.7
1.8
Ž
.
in 5 years percent
Cumulative Prepayment 57.0
Ž
.
in 10 years percent
Cumulative Prepayment 73.2
in 15 years percent
Ž
.
Cumulative Default
0.7
1.4
1.6
Ž
.
in 5 years percent
Cumulative Default
in 10 years percent
Cumulative Default
0.2
0.8
2.6
5.1
Ž
.
.
0.2
0.8
2.8
6.1
Ž
in 15 years percent
No. of Observations
22,294
7,420
7,542
4,988
2,344
N
OTE: Standard deviations are in parentheses.
a Mean value of variables at termination or at the end of the observation period.
assume prepayment and default risks are interdependent. However, the models
do not address unobserved heterogeneity.
Model 1 in Table III tests the ‘‘ruthless’’ model, i.e., Model I, as described in
Section 2. The model includes only measures of the financial value of the
prepayment and default options. The results provide support for the option
theory in that the prepayment hazard increases when the call option is in the
money, and a higher probability of negative equity increases the default hazard
and reduces the prepayment hazard. The results also indicate that the estimated
second order effect of the prepayment option is significant and positive, suggest-
ing that after the interest rate drops below the critical point rU as discussed in
Section 2, the prepayment speed increases substantially.14
14 However, that is not the case in the default function for which the second term is essentially
Ž
.
zero. It is also the case that low interest rates a high value of Call Option lead to negative equity.
We have estimated all the models with Put Option computed from both market and par value of
the mortgages and found no significant changes in any parameter estimates. See Appendix A.

MORTGAGE TERMINATIONS
TABLE III
289
MAXIMUM LIKELIHOOD ESTIMATES FOR COMPETING RISKS OF MORTGAGE
PREPAYMENT AND DEFAULT WITHOUT HETEROGENEITY
Model 1
Prepayment
Model 2
Prepayment
Default
Default
Ž
Call Option fraction of
4.795
124.84
y3.607
6.283
4.837
6.768
.
Ž
.
Ž
Ž
.
Ž
.
Ž
.
contract value
20.10
15.286
118.61
y3.495
19.78
8.662
Ž
Put Option probability
.
Ž
.
.
Ž
.
Ž
.
of negative equity
Squared Term of Call
Option
Squared Term of Put
Option
0.6-LTVF0.75
8.76
2.663
19.79
7.79
2.695
9.19
0.236
y0.359
Ž
.
Ž
.
Ž
.
Ž
.
20.76
3.476
0.32
y16.343
Ž
20.49
3.373
0.20
y9.199
Ž
.
.
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
5.45
14.76
5.00
0.035
7.42
1.384
.
.
1.39
0.060
2.16
2.231
0.75-LTVF0.8
0.8-LTVF0.9
LTV)0.9
.
.
2.69
0.077
3.75
3.146
.
.
3.19
5.36
3.518
y0.056
Ž
.
.
1.89
5.98
0.097
State Unemployment
Rate percent
State Divorce Rate
y0.007
Ž
.
Ž
Ž
.
.
1.37
0.032
3.00
0.415
Ž
.
.
.
percent
LOC
3.10
0.803
4.59
0.059
1.330
2.517
Ž
.
Ž
.
Ž
.
.
1.25
111.42
13.28
18.45
Log Likelihood
y74,981
y74,813
N
OTE: t ratios are in parentheses. All models are estimated by ML approach with flexible baseline hazard
function. Prepayment and default functions are considered as correlated competing risks and they are
estimated jointly. Restrictions of homogeneous error terms were imposed during the maximum likelihood
estimation. LOC is the location parameter of the error term.
Model 2 in Table III extends the ‘‘ruthless’’ model by adding asymmetric
information and the trigger event variables, such as original LTV category,
unemployment, and divorce. The results show that financial motivation is still of
paramount importance governing the prepayment and default behavior. In
addition, the results suggest that borrowers’ willingness to exercise financial
options may be triggered or hindered by other events. For example, it suggests
that higher default risks are associated with higher original LTV’s. This is
Ž
Ž
..
consistent with the argument in Section 2 see Yezer, Phillips, and Trost 1994
that information is asymmetric, and riskier borrowers choose high LTV loans.
The prepayment risk increases slightly as original LTV increases, except for the
highest LTV category. For loans with original LTV over 90 percent, the
prepayment risk is reduced. The estimates also show that unemployment and
divorce are positive and highly significant in the default function, reflecting

290
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
liquidity constraints and the effect of trigger events upon the exercise of put
options.15
Table IV reports the maximum likelihood estimates of the interdependent
competing risks of mortgage prepayment and default with unobserved hetero-
geneity as specified in Section 3. In Models 3 and 4, we assume that there are
two populations among borrowers. The difference in riskiness between these
Ž
two groups may be attributed to unmeasured house-specific factors such as
.
unexpected depreciation or appreciation in property values as well as to
borrower tastes or abilities. Each borrower may either belong to the high risk
group or the low risk group. We do not observe directly the group to which an
individual borrower belongs. Since unobserved heterogeneity may be correlated
with the errors in the competing risks of prepayment and default hazard
functions, we estimate the distribution of the unobserved heterogeneity jointly
with the competing risks of prepayment and default hazard functions.
Model 3 reestimates the ‘‘ruthless’’ model reported in Table III. The esti-
mates still provide support for the predictions of option theory: the prepayment
hazard increases when the call option is in the money; similarly a higher
probability of negative equity increases the default hazard and reduces the
prepayment hazard. However, the marginal effect of the prepayment option,
‘‘Call Option,’’ reported in Table IV increases substantiallyᎏby about 20
percent compared to that reported in Table III. This suggests that estimating
the prepayment risk without accounting for heterogeneity leads to a substantial
underestimate of option-driven prepayment behavior. The estimates also show
that there is a substantial and statistically significant difference between the two
groups in exercising the prepayment option. The borrowers in the high risk
group are about 4.73 times riskier than the borrowers in the low risk group in
Ž
.
terms of prepayment risks 1.972r0.417 . However, there is almost no difference
Ž
.
between the two groups in terms of default risks 2.577r2.403 .
Model 4 reestimates Model 2 reported in Table III. In general, the impor-
tance of the option values reported in Model 3 is confirmed. In addition, the
unemployment variable is negative and highly significant in the prepayment
Ž
functionᎏindicating that liquidity constraints which make refinancing more
difficult for unemployed and divorced households keep them from exercising
.
in-the-money call options. It seems clear that the original LTV, unemployment,
and divorce may trigger or hinder the borrower’s willingness to exercise the
options. These findings are analogous to those noted in Model 2 in Table III.
Note that, in Model 4, after including trigger event variables explicitly in the
prepayment and default hazard function, the estimated heterogeneity becomes
less significant relative to its importance in Model 3. Nonetheless, we still find
15 We also added the difference between the coupon on the mortgage and the average coupon
Ž
.
rate during the quarter in which the loan was originated to test following the argument in Section 2
whether high coupon loans prepay faster andror default more. We found both signs to be positive,
but not statistically significant; as a result we do not include relative coupon rate in any of the results
presented in the tables.

MORTGAGE TERMINATIONS
TABLE IV
291
MAXIMUM LIKELIHOOD ESTIMATES FOR COMPETING RISKS OF MORTGAGE PREPAYMENT AND
DEFAULT WITH UNOBSERVED HETEROGENEITY
Model 3
Model 4
Model 5
Prepay Default
Model 6
Prepay
Default
Prepay
Default
Prepay
Default
Ž
Call Option fraction
5.810
6.324
16.40
5.779
88.51
6.749
17.06
5.921
83.60
6.550
17.90
5.891
81.73
6.877
.
Ž
.
Ž
Ž
.
Ž
.
Ž
.
Ž
.
Ž
Ž
.
Ž
.
Ž
.
of contract value
89.90
y4.700
18.02
9.415
Put Option
Ž
15.275 y4.312
8.670 y5.110
16.891 y4.501
Ž
.
.
Ž
.
Ž
.
Ž
.
.
Ž
.
Ž .
9.30
probability
of negative
10.06
19.82
8.53
9.24
10.61
19.05
8.60
.
equity
Squared Term of
Call Option
4.076 y0.269
4.133
23.96
0.194
0.16
4.135
23.98
1.057
0.88
4.242
23.98
1.032
0.81
Ž
.
Ž
.
Ž
.
Ž
.
Ž
.
Ž
.
Ž
.
Ž
.
23.96
0.23
Squared Term of
Put Option
0.6-LTVF0.75
4.656 y16.333
4.268 y9.207
5.146 y17.760
4.539 y9.882
Ž
.
Ž
.
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
.
Ž
.
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
6.83
14.70
5.94
0.019
7.44
1.385
7.40
14.66
6.17
0.018
7.65
1.351
.
.
.
.
0.60
0.060
2.15
2.230
0.51
0.053
2.10
2.194
0.75-LTVF0.8
0.8-LTVF0.9
LTV)0.9
.
.
.
.
2.06
0.078
3.73
3.146
1.74
0.068
3.67
3.103
.
.
.
.
2.46
5.33
3.516
2.05
5.25
3.471
y0.011
y0.026
Ž
.
.
Ž
.
.
0.29
y0.029
5.94
0.097
0.65
y0.036
5.85
0.103
State Unemploy-
ment Rate
Ž
.
.
Ž
.
.
5.49
2.89
6.36
3.03
Ž
.
percent
State Divorce Rate
y0.005
0.416
y0.016
0.424
Ž
.
Ž
.
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
.
percent
0.43
1.696
4.55
0.058
1.20
2.619
4.57
0.015
LOC1
LOC2
LOC3
1.972
2.577
2.803
0.036
Ž
Ž
.
Ž
.
Ž
Ž
.
.
Ž
Ž
.
Ž
Ž
Ž
.
.
.
37.60
0.417
7.43
2.402
Ž
12.95
0.370
1.15
0.060
11.46
0.972
0.09
3.805
7.82
0.859
0.58
0.080
.
.
.
.
.
.
.
.
15.23
4.21
11.23
1.25
10.42
0.135
9.23
0.221
7.72
0.116
1.24
0.009
Ž
.
.
.
.
0.73
3.70
1.03
3.19
MASS2
MASS3
0.335
10.28
0.379
10.26
1.304
3.69
1.252
Ž
.
Ž
.
Ž
.
Ž
.
3.84
0.119
0.132
Ž
.
Ž
.
2.85
2.55
Log Likelihood
y74,708
y74,560
y74,673
y74,530
N
OTE: t ratios are in parentheses. All models are estimated by ML approach with flexible baseline hazard function.
Prepayment and default functions are considered as correlated competing risks and they are estimated jointly. A bivariate
distribution of unobserved heterogeneous error terms is also estimated simultaneously with the competing risks hazard
functions. LOC1, LOC2, and LOC3 are the location parameters of the error distribution. MASS1, MASS2, and MASS3 are the
mass points associated with LOC1, LOC2, and LOC3, respectively. MASS1 is normalized to 1.0 during the estimation.
that unobserved heterogeneity is statistically significant in exercising the prepay-
ment option, and the borrowers in the high risk group are about 4.58 times
riskier than the borrowers in the low risk group in terms of prepayment risks.
The results also show that by adding trigger event variables, Model 4 has a much
better fit than Model 3, the ‘‘ruthless’’ model.

292
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
Table IV also reports estimates of two models that extend the Models 3 and 4
by allowing three mass points in the distribution of the unobserved hetero-
geneities. There are no significant effects on the estimation of the explicitly-
specified explanatory variables. However, the three estimated location variables
are significantly different from each other, and all the estimated mass points are
significant in the prepayment function. The log likelihood values also suggest
that Model 5 and Model 6 provide better fit to the data than Models 3 and 4,
respectively.
The estimates from both Models 5 and 6 suggest that over forty percent of the
borrowers are in the group most likely to exercise the mortgage prepayment
Ž
w
x.
option e.g., 1r 1q1.252q0.119 . About 5 percent of the borrowers are much
Ž
w
slower in exercising the mortgage prepayment option e.g., 0.119r 1q1.252q
0.119 . The behavior of the remaining half of the borrowers is somewhere in
between these two extreme groups. The estimates also suggest that given the
same market and economic environment i.e., given the market interest rate,
x.
Ž
pattern of house price appreciationrdepreciation rate, unemployment, and
.
divorce rates, etc. , the high risk group is about three times riskier than the
intermediate group, and about twenty times riskier than the low risk group in
terms of prepayment. However, the results also show that heterogeneity is less
important among borrowers in terms of exercising the default option.16
Figure 2 graphs the average conditional prepayment and default rates by
mortgage age estimated by Model 6 for the three groups. Panel A indicates that
conditional prepayment rates increase from about 2 percent, on average, after 5
years, to about 6 percent after 8 years. However, the average masks quite
different behavior among the three groups. For the middle group, the estimated
prepayment rate increases from about 1.5 percent to 4 percent, while for the
Ž
.
third low risk group the conditional prepayment rate changes only a little.
Ž
.
Among the first the high risk group, however, conditional prepayment rates
increase from about 3 percent after 5 years to about 11 percent after 8 years.
Panel B reports the average conditional default rates estimated for the three
groups. The panel confirms that inverted U shaped function implied by theory
Ž
Ž
..
e.g., Quigley and Van Order 1995 , but again there are large behavioral
differences among groups. At the peak, after about 6 years, conditional default
rates are less than 0.02 percent for those in group 1 and about 5 times as large
for those in group 3. Unobserved heterogeneity matters a lot in the behavior of
mortgage holders.
16 We have sought to extend the models reported in Table IV to estimate additional mass points
in the distribution of unobserved heterogeneity. When four mass points are estimated, the value of
Ž
.
the log likelihood function is unchanged to five significant digits , and several of the estimated
location and mass points are insignificant. The fraction of borrowers classified into the high risk
Ž
.
extreme 41 percent is about the same; the fraction classified into the low risk extreme is about zero
but the coefficient on the mass point is insignificant . When five or six mass points are estimated,
Ž
.
the coefficients are insignificant. In all these estimations, however, the coefficients of the option
variables and the other parameters are virtually unchanged.

MORTGAGE TERMINATIONS
293
FIGURE 2
Table V reports the results of estimating the mortgage prepayment risk
and default risk functions independently using three and two mass points
Ž
.
respectively the best fitting models . The qualitative pattern of the estimated
coefficients is similar. However, it is quite clear that, for this sample at least,
ignoring the interdependence between prepayment risk and default risk has a

294
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
substantial effect upon the accuracy of the estimation, especially for the default
function.17
Figure 3 summarizes the predictions of the various specifications of the
competing risks models, indicating the conditional prepayment and default rates
Ž
.
Ž
predicted by Model 2 interdependent risks, no heterogeneity , Model 5 fric-
.
Ž
tionless model, interdependent risks, heterogeneity , Model 6 interdependent
.
Ž
.
risks, heterogeneity , and Model 8 independent risks, heterogeneity . Again the
mean value of the simulated conditional prepayment and default rates are
presented here. The predictions of the models are similar, but they are certainly
not identical. The specification preferred on statistical grounds, Model 6, does
yield substantially different predictions.
Figure 4 reports the cumulative prepayment and default rates estimated by
Model 6 for three origination year cohorts: 1978:IV, 1980:IV, and 1982:IV. The
figure also reports the raw unadjusted Kaplan-Meier cumulative rates. The
economic model tracks the raw data well, on average, but there is more volatility
in the unadjusted data than in the predictions of the model. In particular, the
covariates do not capture the rapid increase in termination by prepayment
during 1985᎐1986.
4.2. Regional Variation
In this section we present additional estimates of the interdependent compet-
ing risks model with heterogeneity for two regions, California and Texas. The
California economy was strong and growing throughout most of the sample
period, while the Texas economy was a victim of the oil crisis and a prolonged
depression. For California, we draw a simple random sample of ten percent
Ž
.
Ž
.
22,374 loans out of the population of 222,656 loans. For Texas, we analyze
Ž
.
the full sample of 29,310 loans.
Table VI reports the means and standard deviations of the explanatory
variables in these two states at mortgage origination and at termination. When
Ž
.
all loans are compared at origination with the national sample Table I , the
average value of the call option is very similar, and the initial LTV’s are not very
different. However, at origination the probability of negative equity is only a
third as large for California loans as for the US as a whole, while for Texas the
probability of negative equity is more than twice as large as elsewhere. The
qualitative pattern of the averages for California and Texas, however, are similar
to those reported in Table I for the US as a whole.
Despite these patterns, there are substantial quantitative differences between
Ž
.
the two states. The average value of the call option is larger i.e., less negative
17
Ž
.
Model 7 and Model 8 are estimated based on the same sample of 22,294 loans that are used
for estimating Model 5 and Model 6. Table V reports that for Model 8, the log likelihood values are
Ž
y80,935 and y417,644 for prepayment and default functions, respectively compared to y74,530 in
Model 6 . The large negative value of the log likelihood function in the estimation of the
.
independent default hazard model arises mainly because only 1.6% of the sample are defaulted
loans.

MORTGAGE TERMINATIONS
TABLE V
295
MAXIMUM LIKELIHOOD ESTIMATES FOR INDEPENDENT RISKS OF MORTGAGE PREPAYMENT
AND DEFAULT WITH UNOBSERVED HETEROGENEITY
Model 7
Prepayment
Model 8
Prepayment
Default
Default
Ž
Call Option fraction of
6.002
8.348
5.976
8.520
.
Ž
.
Ž
Ž
.
Ž
.
Ž
.
contract value
83.51
y3.989
21.07
15.743
82.24
y3.622
21.34
8.189
Ž
Put Option probability
.
Ž
.
.
Ž
.
Ž
.
of negative equity
Squared Term of Call
Option
Squared Term of Put
Option
0.6-LTVF0.75
8.37
4.287
17.96
4.242
7.03
4.383
7.83
3.374
Ž
.
Ž
.
Ž
.
Ž
.
24.53
4.260
3.20
y16.803
24.62
3.911
2.57
y9.042
Ž
.
Ž
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
6.12
12.78
5.33
0.020
6.23
1.327
.
.
0.60
0.059
2.07
2.372
0.75-LTVF0.8
0.8-LTVF0.9
LTV)0.9
.
.
1.95
0.083
3.98
3.370
.
.
2.52
0.009
5.74
3.333
.
.
0.22
5.66
0.166
State Unemployment
Rate percent
State Divorce Rate
y0.033
Ž
.
Ž
.
.
5.96
y0.006
5.04
0.390
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
.
.
percent
0.46
2.308
4.24
0.144
LOC1
LOC2
LOC3
MASS2
MASS3
2.630
10.857
Ž
.
Ž
.
.
.
11.92
0.901
7.05
0.506
8.16
0.759
1.23
0.011
Ž
Ž
Ž
Ž
.
Ž
.
.
.
0.99
8.28
0.109
1.97
7.13
0.095
.
.
2.78
1.017
2.73
1.033
0.628
0.344
.
Ž
.
.
Ž
.
3.52
0.100
3.42
3.87
0.099
2.31
.
.
2.39
2.37
Log Likelihood
y80,968
y417,754
y80,935
y417,644
N
OTE: t ratios are in parentheses. All models are estimated by ML approach with flexible baseline hazard
function. Prepayment and default functions are estimated separately. A univariate distribution of unobserved
heterogeneous error terms is also estimated simultaneously with the prepayment or default hazard functions.
LOC1, LOC2, and LOC3 are the location parameters of the error distribution. MASS1, MASS2, and MASS3
are the mass points associated with LOC1, LOC2, and LOC3, respectively. MASS1 is normalized to 1.0 during
the estimation.
in California at origination and at termination for mortgages of all types.
Conversely, the average probability of negative equity is larger in Texas at
origination for mortgages of all types. In part, this reflects the higher average
LTV in Texas.
In both states, the average unemployment rate at termination by default is
higher than at termination by prepayment, but in Texas the average unemploy-
ment rate at termination is almost forty percent higher than at origination.

296
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
FIGURE 3
Differences between the two states in the averages of the variable measuring the
value of the put option are particularly striking. In California, for those
mortgages terminating by default, the probability of negative equity at origina-
tion is about 2 percent; the probability of negative equity at termination is about
2.5 percent. In Texas, however, for defaulted mortgages, the probability of
Ž
negative equity is more than 5 percent at origination raising real questions
about the oversight provided by lending organizations, mostly savings and loan

MORTGAGE TERMINATIONS
297
FIGURE 4

298
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
Ž
.
Continued
FIGURE 4

MORTGAGE TERMINATIONS
TABLE VI
299
DESCRIPTIVE STATISTICS ON CALIFORNIA AND TEXAS MORTGAGE LOANS
MEAN VALUES AT ORIGINATION AND TERMINATION
At Origination
At Termination
Variable
All Loans
Prepaid
Defaulted
Othera
Prepaid
Defaulted
A. California
Call Option fraction of y0.0505
contract value
Ž
y0.0526
y0.0628
y0.0414
y0.0078
0.1000
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
.
0.0055
0.0027
0.0060
0.0027
0.0095
0.0033
0.0293
0.0214
0.0246
0.0012
7.0468
Ž
Put Option probability
0.0194
0.0013
7.4231
0.0013
0.0001
7.3015
0.0026
0.0001
6.6792
.
.
.
.
.
.
.
of negative equity
State Unemployment
0.0002
7.3038
0.0002
7.3018
Ž
.
.
.
.
.
.
.
Rate percent
State Divorce Rate
1.4484
5.7825
1.5275
5.7652
1.6332
5.5526
1.1215
5.8712
1.7284
5.0259
1.3618
5.1504
Ž
.
.
.
.
.
.
.
percent
Initial Loan-To-Value
0.0534
0.7581
0.0552
0.7575
0.0310
0.8655
0.0222
0.7514
0.2258
ᎏ
0.0785
ᎏ
Ž
.
.
.
.
.
0.0203
Ratio LTV
0.0213
0.0217
0.0056
No. of Observations
22,374
17,598
359
4,417
17,598
359
B. Texas
Call Option fraction of y0.0616
Ž
y0.0631
y0.0889
y0.0552
y0.0680
0.0364
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
.
contract value
0.0095
0.0159
0.0098
0.0185
0.0531
0.0074
0.0403
0.0178
0.2690
0.0879
7.3154
Ž
Put Option probability
0.0119
0.0008
5.2463
0.0151
0.0009
4.9635
0.0507
0.0212
7.2736
.
.
.
.
.
.
.
of negative equity
State Unemployment
Rate percent
State Divorce Rate
percent
Initial Loan-To-Value
Ratio LTV
0.0010
5.1320
0.0028
5.2709
Ž
.
.
.
.
.
.
.
0.7868
6.4923
0.9558
6.5083
1.4325
6.5674
0.4193
6.4556
1.6342
5.9555
1.0645
5.7162
Ž
.
.
.
.
.
.
.
0.0384
0.8007
0.0584
0.7828
0.0543
0.9086
0.0519
0.8063
0.0923
ᎏ
0.0583
ᎏ
Ž
.
.
.
.
.
0.0265
0.0267
0.0274
0.0084
No. of Observations
29,310
15,364
1,919
12,027
15,364
1,919
N
OTE: Standard deviations are in parentheses.
a ‘‘Other’’ includes matured mortgages as well as those outstanding at the end of the observation period.
.
institutions, during the period . At termination, the probability of negative
equity is almost 27 percent.
Table VII reports the coefficient estimates of the behavioral model for
California and Texas. The model is estimated to include a bivariate distribution
of unobserved heterogeneous error terms.18 For convenience the relevant model
from Table IV for the US as a whole is reproduced.
With only three exceptions involving insignificant coefficients, the pattern of
the estimated coefficients is the same for the two states, but the magnitudes of
the estimates vary. Each of the key financial variables exerts a substantial direct
18Again, attempts to estimate three or more mass points failed. For California, the maximum
likelihood estimation with three mass points yields imprecise estimates, and the log likelihood ratio
remains the same as in Table VII. For Texas, the estimation with 3 mass points does not converge at
all.

300
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
TABLE VII
a
COMPARISON OF PREPAYMENT AND DEFAULT RATES AMONG US, CALIFORNIA, AND TEXAS LOANS
US
Prepayment
Califomia
Prepayment Default
Texas
Prepayment
Default
Default
Ž
Call Option fraction of
5.779
6.750
5.769
9.081
5.269
68.22
7.503
.
Ž
.
Ž
.
Ž
.
Ž
.
Ž
.
Ž
Ž
.
contract value
88.51
y4.312
17.06
8.669
97.84
y11.837
21.39
20.872
27.92
4.892
Ž
Put Option probability
y1.917
.
Ž
.
Ž
.
Ž
.
Ž
.
Ž
Ž
.
.
of negative equity
Squared Term of Call
Option
Squared Term of Put
Option
0.6-LTVF0.75
8.53
4.133
9.24
0.193
0.15
8.58
3.568
4.73
1.869
10.80
3.421
15.24
1.430
Ž
.
Ž
.
Ž
.
Ž
.
.
Ž
.
23.96
4.268
19.11
1.34
20.61
1.791
1.57
y2.676
y9.206
45.372
y92.270
Ž
Ž
Ž
Ž
.
Ž
Ž
Ž
Ž
Ž
Ž
Ž
Ž
Ž
.
Ž
.
Ž
Ž
Ž
Ž
Ž
.
Ž
.
Ž
Ž
Ž
.
5.94
0.019
7.44
1.385
4.74
0.057
3.33
2.802
7.68
0.046
6.78
1.136
.
.
Ž
.
.
Ž
.
.
0.60
0.060
2.15
2.230
2.01
3.84
2.955
1.39
5.45
1.493
0.75-LTVF0.8
0.8-LTVF0.9
LTV)0.9
y0.006
y0.005
.
.
Ž
Ž
Ž
.
.
Ž
.
.
2.06
0.078
3.73
3.146
0.24
0.074
4.12
3.747
0.17
y0.043
8.26
2.045
.
.
.
.
Ž
.
Ž
Ž
.
2.46
5.33
3.517
2.65
0.023
5.23
4.193
1.37
y0.322
11.91
2.759
y0.011
Ž
.
.
.
.
Ž
.
.
0.29
y0.029
5.94
0.097
0.45
5.77
10.43
0.011
16.59
State Unemployment
Rate percent
State Divorce Rate
y0.017
y0.151
y0.115
Ž
.
Ž
.
.
Ž
.
Ž
Ž
Ž
Ž
.
Ž
.
Ž
Ž
Ž
Ž
.
5.49
y0.005
2.89
0.416
2.73
0.527
2.47
0.467
1.48
0.368
4.47
0.265
Ž
.
Ž
.
.
Ž
.
.
Ž
.
.
percent
0.43
1.696
4.55
0.058
23.34
0.106
1.89
0.035
11.05
0.104
2.52
0.717
LOC1
Ž
Ž
.
.
Ž
.
.
Ž
.
.
12.95
0.370
1.15
0.060
8.00
0.020
0.70
0.001
4.38
0.021
1.56
0.036
LOC2
.
.
Ž
.
.
Ž
.
.
1.46
11.23
1.25
3.54
0.30
4.04
MASS2
0.379
10.26
0.061
3.44
0.353
7.90
Ž
.
Ž
.
Ž
.
Log Likelihood
y74,560
y77,290
y87,028
N
OTE: t ratios are in parentheses. All models are estimated by ML approach with flexible baseline hazard function.
Prepayment and default functions are considered as correlated competing risks and they are estimated jointly. A bivariate
distribution of unobserved heterogeneous error terms is also estimated simultaneously with the competing risks hazard
functions. LOC1 and LOC2 are the location parameters of the error distribution. MASS1 and MASS2 are the mass points
associated with LOC1 and LOC2, respectively. MASS1 is normalized to 1.0 during the estimation.
a The US sample is based on a five percent random sample of 22,294 loans. The California sample is based on a ten
percent random sample of 22,374 loans. The Texas sample is based on a hundred percent sample of 29,310 loans.
effect upon the exercise of both options. For Texas loans, in contrast to
California loans or US loans in general, there appears to be substantial hetero-
Ž
geneity in default behavior however, the t statistics associated with LOC1 and
.
LOC2 are barely significant . Individuals in the first group are almost 20 times
as likely to default, ceteris paribus, than those classified in the second group.
This difference is not precisely estimated, but it suggests substantially different
behavior among Texas borrowers.
Figure 5 presents the prepayment and default rates for California and Texas
estimated using the separate models reported in Table VII. Panel A reports the
prepayment rates, illustrating the higher conditional rates in California after

MORTGAGE TERMINATIONS
301
FIGURE 5

302
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
Ž
.
Continued
FIGURE 5

MORTGAGE TERMINATIONS
303
about six years, while Panel B illustrates the much higher conditional default
rates predicted in Texas after about six years. Panels C and D report the
simulated average prepayment and default rates using the coefficients from the
Ž
.
national model columns 1 and 2 of Table VII and the realizations in the data
for California and Texas. The projected differences for the two states are
qualitatively similar, but the national model understates the divergence in the
behavior of mortgage holders in the two states. Regional differences matter a
lot, even beyond the differences in the course of housing and labor markets.
5. CONCLUSION
This paper has presented a unified model of the competing risks of mortgage
termination by prepayment and default. The model considers these two hazards
as dependent competing risks and estimates them jointly. The model also
accounts for the unobserved heterogeneity among borrowers, and estimates the
unobserved heterogeneity simultaneously with the parameters and baseline
hazards associated with prepayment and default functions.
The substantive results of the analysis provide support for the contingent
claims model, which predicts the exercise of financial options. The financial
value of the call option is strongly associated with exercise of the prepayment
option, and the probability that the put option is in the money is strongly
associated with exercise of the default option. The results also provide strong
support for the interdependence of the decisions to prepay and to default on
mortgage obligations.
The results also show that there exists significant heterogeneity among mort-
gage borrowers, particularly regarding prepayment. The results indicate that
ignoring heterogeneity among mortgage borrowers leads to serious errors in
estimating the prepayment risk. Moreover, forecasts that ignore the interdepen-
dence between default and prepayment risks and that estimate these two risks
separately lead to serious errors in estimating the default risk.
The results also point to differences in prepayment and default behavior
across regions, arising from variations in institutions or behavioral responses as
well as variations in market conditions.
Further, the results suggest that, holding other things constant, those who
have chosen high initial LTV loans are more likely to exercise options in the
mortgage marketᎏprepayment as well as default. It appears that the initial
LTV ratio, known at the time mortgages are issued, may well reflect investor
preferences for risk in the market for mortgages on owner-occupied housing.
Finally, unemployment and divorce rates have significant effects on default.
Taken together, all these results suggest that the simple option model is not
enough.

304
Y. DENG, J. M. QUIGLEY, AND R. VAN ORDER
Uni¨ersity of Southern California, Los Angeles, CA 90089-0626, U.S.A.;
ydeng@usc.edu,
Uni¨ersity of California, Berkeley, CA 94720-3880, U.S.A.; quigley@econ.berke-
ley.edu,
and
Freddie Mac, 8200 Jones Branch Dri¨e, McLean, VA 22102, U.S.A.;
order ¨an@freddiemac.com.
᎐
Manuscript recei¨ed May, 1995; final re¨ision recei¨ed February, 1999.
APPENDIX A: SPECIFICATIONS OF ‘‘CALL OPTION’’ AND ‘‘PUT OPTION’’ VARIABLES
The variables measuring the value of the put and call options are defined by the initial terms of
the mortgage and current market conditions. For fixed-rate level-payment mortgage i with an
original loan amount of Oi, a mortgage note rate of ri, and a monthly payment of Pi, in principal
and interest, the mortgage term in quarters, TMi, is
Ž
.
Pi yOi = rir1200
log
ž
/
Pi
Ž
.
10
TMi s
.
1
log
=3
ž1qr r1200
/
i
At each quarter k after origination at time ␶i, the local market interest rate is mj, ␶ qk , where j
indexes the local region. The ‘‘Call Option’’ variable is defined as the difference in i the present
values of the payment stream at the mortgage note rate and the current interest rate:
TMiyki
TMiyki
Pi =3
Pi =3
t y
Ý
Ý
t
Ž
.
Ž
.
1qrir400
1qmj, ␶ qk r400
t
s1
ts1
i
i
Ž
.
11
Call Optioni, k
s
s
TMiyki
i
Pi =3
1qmj, ␶ qk r400
Ý
t
Ž
.
ts1
i
i
Vi, m j, ␶ qk yViU, r
i
i
i
.
Vi, m j, ␶ qk
i
i
As shown in Appendix B, the market value Mi of property i, purchased at a cost of Ci at time ␶
i
and evaluated ki quarters thereafter is
Ij, ␶
kq
i
i
Ž
.
12
Mi, k sCi
i
ž /
Ij, ␶
i
where the term in parentheses follows a log normal distribution.
The ratio of equity to market value, E, of the property i is
Mi, k yVi, m j, ␶ qk
i
i
i
Ž
.
13
Ei, k
s
.
i
Mi, k
i